ESCs use digital microprossesers and metalic oxide field effect transistors (MOSFETs) to control the voltage and phase sequence sent to a brushless motor. The power to the motor is controlled by rapidly pulsing the voltage on and off at a very high frequency. For a brushed DC motor, controlling the power is all the ESC does. For a brushless motor, the ESC also inverts the DC into a square wave three phase alternating current.
Most ESCs also have a voltage regulator that lowers the battery voltage to 4.8 volts to operate the radio reciever and servos. This is called a BEC or battery eliminator circuit. If you use more than a certain number of servos or use more than a certain number of cells, you may not be able to use the built in BEC and will have to operate the radio and servos with a separate battery like glow planes do.

The amp rating of your ESC needs to be at least as high as the motor is going to draw. Oversizing the ESC does no harm other than making the plane needlessly heavy.